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An approximate universal coefficient theorem

Huaxin Lin
2005 Transactions of the American Mathematical Society  
An approximate Universal Coefficient Theorem (AUCT) for certain C * -algebras is established.  ...  We also show that C * -algebras that are locally approximated by C * -algebras satisfying the AUCT satisfy the AUCT.  ...  Part of this work was done when the author was visiting East China Normal University. He thanks the Department of Mathematics for providing a shelter during the summer 2000 in Shanghai.  ... 
doi:10.1090/s0002-9947-05-03696-2 fatcat:jr5272drsnemtm7uqmw4tyvn7m

Universal Approximation Theorem for Neural Networks [article]

Takato Nishijima
2021 arXiv   pre-print
Is there any theoretical guarantee for the approximation ability of neural networks? The answer to this question is the "Universal Approximation Theorem for Neural Networks".  ...  This paper is a comprehensive explanation of the universal approximation theorem for feedforward neural networks, its approximation rate problem (the relation between the number of intermediate units and  ... 
arXiv:2102.10993v1 fatcat:i3rbo7bs2jgvzlwhieapetitdi

Universal approximation theorem for Dirichlet series

O. Demanze, A. Mouze
2006 International Journal of Mathematics and Mathematical Sciences  
Several density results are proved that finally lead to the main theorem on simultaneous approximation.  ...  The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right  ...  The present paper can also be seen as an extension of [6] to universal Dirichlet series. Moreover, Theorem 4.3 is a universal approximation result in the sense of [7] .  ... 
doi:10.1155/ijmms/2006/37014 fatcat:737d3r4ktzdxrb4547e3jz6qt4

Mergelyan's approximation theorem with nonvanishing polynomials and universality of zeta-functions

Johan Andersson
2013 Journal of Approximation Theory  
We prove a variant of the Mergelyan approximation theorem that allows us to approximate functions that are analytic and nonvanishing in the interior of a compact set K with connected complement, and whose  ...  We apply this result on the Voronin universality theorem for compact sets K of this type, where the usual condition that the function is nonvanishing on the boundary can be removed.  ...  By Conjecture 2 we can approximate f (z) by a polynomial p(z) such that Relating Mergelyan's theorem and Voronin universality |p(z) − f (z)| < ε/2, (z ∈ K), (1) where p(z) is nonvanishing on K.  ... 
doi:10.1016/j.jat.2012.12.005 fatcat:zelqzp3lfnakbfooevr6lvs5pq

A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributions [article]

Yulong Lu, Jianfeng Lu
2020 arXiv   pre-print
This paper studies the universal approximation property of deep neural networks for representing probability distributions.  ...  We prove upper bounds for the size (width and depth) of the deep neural network in terms of the dimension d and the approximation error ε with respect to the three discrepancies.  ...  Our main result is the universal approximation theorem for expressing probability distributions. Theorem 2.1 (Main theorem).  ... 
arXiv:2004.08867v3 fatcat:bassh6lnu5czljl746ratj3eey

Universal Approximation Theorems for Differentiable Geometric Deep Learning [article]

Anastasis Kratsios, Leonie Papon
2022 arXiv   pre-print
In the Euclidean setting, our results imply a quantitative version of Kidger and Lyons (2020)'s approximation theorem and a data-dependent version of Yarotsky and Zhevnerchuk (2019)'s uncursed approximation  ...  As applications, we confirm the universal approximation capabilities of the following GDL models: Ganea et al. (2018)'s hyperbolic feedforward networks, the architecture implementing Krishnan et al. (2015  ...  These networks are obtained via the universal approximation theorem. Hence, to derive the estimates of Proposition 53, we should constructively prove the universal approximation theorem (Theorem 46).  ... 
arXiv:2101.05390v4 fatcat:dgbgiv7ymrbttapiwlgiizrtaa

The universal approximation theorem for complex-valued neural networks [article]

Felix Voigtlaender
2020 arXiv   pre-print
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks.  ...  We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function  ...  Related work The classical universal approximation theorem There exist many versions of the universal approximation theorem for real networks.  ... 
arXiv:2012.03351v1 fatcat:3wq474t47vctre3oidrbpugo6a

Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks [article]

Annan Yu, Chloé Becquey, Diana Halikias, Matthew Esmaili Mallory, Alex Townsend
2021 arXiv   pre-print
The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth.  ...  Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators.  ...  In the approximation theory of neural networks (NNs), universal approximation theorems (UATs) are statements that establish the density of a class of NNs within a space of mappings.  ... 
arXiv:2109.11354v1 fatcat:rtezxzkrubglzikwjikbgqoeji

A Universal Approximation Theorem for Mixture-of-Experts Models

Hien D. Nguyen, Luke R. Lloyd-Jones, Geoffrey J. McLachlan
2016 Neural Computation  
Our result can be viewed as a universal approximation theorem for MoE models.  ...  The theorem we present allows MoE users to be confident in applying such models for estimation when data arise from nonlinear and nondifferentiable generative processes.  ...  Our result is a universal approximation theorem, similar in spirit to Cybenko (1989, theorem 2) , where the linear combination of sigmoidal functions is proved dense in C(X).  ... 
doi:10.1162/neco_a_00892 pmid:27626962 fatcat:7gwgc5ahezgenmcftmjyl4dxca

Universal approximation theorem for uninorm-based fuzzy systems modeling

Ronald R. Yager, Vladik Kreinovich
2003 Fuzzy sets and systems (Print)  
doi:10.1016/s0165-0114(02)00521-3 fatcat:ianplle74be4tkdjpslukinfry

A Universal Approximation Theorem for Mixture of Experts Models [article]

Hien D Nguyen, Luke R Lloyd-Jones, Geoffrey J McLachlan
2016 arXiv   pre-print
Our result can be viewed as a universal approximation theorem for MoE models.  ...  Our result is a universal approximation theorem for MoE mean functions in the style of [3] .  ...  Stone-Weierstrass Theorem Following the presentation of [2] , the Stone-Weierstrass Theorem can be phrased as follows. Theorem 1.  ... 
arXiv:1602.03683v1 fatcat:svv6rphry5fdjfj2rfqlujjjo4

Lavrentiev's approximation theorem with nonvanishing polynomials and universality of zeta-functions [article]

Johan Andersson
2010 arXiv   pre-print
We prove a variant of the Lavrentiev's approximation theorem that allows us to approximate a continuous function on a compact set K in C without interior points and with connected complement, with polynomial  ...  We use this result to obtain a version of the Voronin universality theorem for compact sets K, without interior points and with connected complement where it is sufficient that the function is continuous  ...  By Theorem 1 we can approximate f (z) by a polynomial g(z) such that |g(z) − f (z)| < ǫ/2, z ∈ K, (3) where g(z) is nonvanishing on K.  ... 
arXiv:1010.0386v1 fatcat:e3rwnxjmpnekphxsjajhflbdmi

Characterizing the Universal Approximation Property [article]

Anastasis Kratsios
2020 arXiv   pre-print
Moreover, we show that most function spaces admit universal approximators built using a single function.  ...  To better understand the approximation capabilities of various currently available neural network architectures, this paper studies the universal approximation property itself across a broad scope of function  ...  Theorem 2.3 (Equivalence of Universal Approximators to the Feed-Forward Architecture).  ... 
arXiv:1910.03344v3 fatcat:j3mhfpj3mbf35iz2ypotvyt5de

DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators [article]

Lu Lu, Pengzhan Jin, George Em Karniadakis
2020 arXiv   pre-print
This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data.  ...  While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can  ...  Acknowledgments We thank Yanhui Su of Fuzhou University for the help on Theorem 2. We thank Zhongqiang Zhang of Worcester Polytechnic Institute for the proof in Appendix C.  ... 
arXiv:1910.03193v3 fatcat:67kretzwczffriihix3zn7fs6i

A Gradient Free Neural Network Framework Based on Universal Approximation Theorem [article]

Nikolaos P. Bakas, Andreas Langousis, Mihalis Nicolaou, Savvas A. Chatzichristofis
2020 arXiv   pre-print
We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the Universal Approximation Theorem, avoiding laborious iterations.  ...  The method is based on the calculation of the weights of each neuron in small neighborhoods of the data, such that the corresponding local approximation matrix is invertible.  ...  This complies with the Universal Approximation theorem, and offers a geometric point of view.  ... 
arXiv:1909.13563v3 fatcat:5u34jsdskvho7e5bc2fj3u6rai
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